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Fish-eye lens may entangle pairs of atoms



Nearly 150 years ago, the physicist James Maxwell proposed that a circular lens that is thickest at its center, and that gradually thins out at its edges, should exhibit some fascinating optical behavior. Namely, when light is shone through such a lens, it should travel around in perfect circles, creating highly unusual, curved paths of light.

He also noted that such a lens, at least broadly speaking, resembles the eye of a fish. The lens configuration he devised has since been known in physics as Maxwell’s fish-eye lens — a theoretical construct that is only slightly similar to commercially available fish-eye lenses for cameras and telescopes. 

Now scientists at MIT and Harvard University have for the first time studied this unique, theoretical lens from a quantum mechanical perspective, to see how individual atoms and photons may behave within the lens. In a study published Wednesday in Physical Review A, they report that the unique configuration of the fish-eye lens enables it to guide single photons through the lens, in such a way as to entangle pairs of atoms, even over relatively long distances.

Entanglement is a quantum phenomenon in which the properties of one particle are linked, or correlated, with those of another particle, even over vast distances. The team’s findings suggest that fish-eye lenses may be a promising vehicle for entangling atoms and other quantum bits, which are the necessary building blocks for designing quantum computers.

“We found that the fish-eye lens has something that no other two-dimensional device has, which is maintaining this entangling ability over large distances, not just for two atoms, but for multiple pairs of distant atoms,” says first author Janos Perczel, a graduate student in MIT’s Department of Physics. “Entanglement and connecting these various quantum bits can be really the name of the game in making a push forward and trying to find applications of quantum mechanics.”

The team also found that the fish-eye lens, contrary to recent claims, does not produce a perfect image. Scientists have thought that Maxwell’s fish-eye may be a candidate for a “perfect lens” — a lens that can go beyond the diffraction limit, meaning that it can focus light to a point that is smaller than the light’s own wavelength. This perfect imaging, scientist predict, should produce an image with essentially unlimited resolution and extreme clarity.

However, by modeling the behavior of photons through a simulated fish-eye lens, at the quantum level, Perczel and his colleagues concluded that it cannot produce a perfect image, as originally predicted.

“This tells you that there are these limits in physics that are really difficult to break,” Perczel says. “Even in this system, which seemed to be a perfect candidate, this limit seems to be obeyed. Perhaps perfect imaging may still be possible with the fish eye in some other, more complicated way, but not as originally proposed.”

Perczel’s co-authors on the paper are Peter Komar and Mikhail Lukin from Harvard University.

A circular path

Maxwell was the first to realize that light is able to travel in perfect circles within the fish-eye lens because the density of the lens changes, with material being thickest at the middle and gradually thinning out toward the edges. The denser a material, the slower light moves through it. This explains the optical effect when a straw is placed in a glass half full of water. Because the water is so much denser than the air above it, light suddenly moves more slowly, bending as it travels through water and creating an image that looks as if the straw is disjointed.

In the theoretical fish-eye lens, the differences in density are much more gradual and are distributed in a circular pattern, in such a way that it curves rather bends light, guiding light in perfect circles within the lens.

In 2009, Ulf Leonhardt, a physicist at the Weizmann Institute of Science in Israel was studying the optical properties of Maxwell’s fish-eye lens and observed that, when photons are released through the lens from a single point source, the light travels in perfect circles through the lens and collects at a single point at the opposite end, with very little loss of light.

“None of the light rays wander off in unwanted directions,” Perczel says. “Everything follows a perfect trajectory, and all the light will meet at the same time at the same spot.”

Leonhardt, in reporting his results, made a brief mention as to whether the fish-eye lens’ single-point focus might be useful in precisely entangling pairs of atoms at opposite ends of the lens.

“Mikhail [Lukin] asked him whether he had worked out the answer, and he said he hadn’t,” Perczel says. “That’s how we started this project and started digging deeper into how well this entangling operation works within the fish-eye lens.”

Playing photon ping-pong

To investigate the quantum potential of the fish-eye lens, the researchers modeled the lens as the simplest possible system, consisting of two atoms, one at either end of a two-dimensional fish-eye lens, and a single photon, aimed at the first atom. Using established equations of quantum mechanics, the team tracked the photon at any given point in time as it traveled through the lens, and calculated the state of both atoms and their energy levels through time.

They found that when a single photon is shone through the lens, it is temporarily absorbed by an atom at one end of the lens. It then circles through the lens, to the second atom at the precise opposite end of the lens. This second atom momentarily absorbs the photon before sending it back through the lens, where the light collects precisely back on the first atom.

“The photon is bounced back and forth, and the atoms are basically playing ping pong,” Perczel says. “Initially only one of the atoms has the photon, and then the other one. But between these two extremes, there’s a point where both of them kind of have it. It’s this mind-blowing quantum mechanics idea of entanglement, where the photon is completely shared equally between the two atoms.”

Perczel says that the photon is able to entangle the atoms because of the unique geometry of the fish-eye lens. The lens’ density is distributed in such a way that it guides light in a perfectly circular pattern and can cause even a single photon to bounce back and forth between two precise points along a circular path.

“If the photon just flew away in all directions, there wouldn’t be any entanglement,” Perczel says. “But the fish-eye gives this total control over the light rays, so you have an entangled system over long distances, which is a precious quantum system that you can use.”

As they increased the size of the fish-eye lens in their model, the atoms remained entangled, even over relatively large distances of tens of microns. They also observed that, even if some light escaped the lens, the atoms were able to share enough of a photon’s energy to remain entangled. Finally, as they placed more pairs of atoms in the lens, opposite to one another, along with corresponding photons, these atoms also became simultaneously entangled.

“You can use the fish eye to entangle multiple pairs of atoms at a time, which is what makes it useful and promising,” Perczel says.

Fishy secrets

In modeling the behavior of photons and atoms in the fish-eye lens, the researchers also found that, as light collected on the opposite end of the lens, it did so within an area that was larger than the wavelength of the photon’s light, meaning that the lens likely cannot produce a perfect image.

“We can precisely ask the question during this photon exchange, what’s the size of the spot to which the photon gets recollected? And we found that it’s comparable to the wavelength of the photon, and not smaller,” Perczel says. “Perfect imaging would imply it would focus on an infinitely sharp spot. However, that is not what our quantum mechanical calculations showed us.”

Going forward, the team hopes to work with experimentalists to test the quantum behaviors they observed in their modeling. In fact, in their paper, the team also briefly proposes a way to design a fish-eye lens for quantum entanglement experiments.

“The fish-eye lens still has its secrets, and remarkable physics buried in it,” Perczel says. “But now it’s making an appearance in quantum technologies where it turns out this lens could be really useful for entangling distant quantum bits, which is the basic building block for building any useful quantum computer or quantum information processing device.”



Sense, sensibility, and superconductors



Jonathan Monroe disagreed with his PhD supervisor—with respect. They needed to measure a superconducting qubit, a tiny circuit in which current can flow forever. The qubit emits light, which carries information about the qubit’s state. Jonathan and Kater intensify the light using an amplifier. They’d fabricated many amplifiers, but none had worked. Jonathan suggested changing their strategy—with a politeness to which Emily Post couldn’t have objected. Jonathan’s supervisor, Kater Murch, suggested repeating the protocol they’d performed many times.

“That’s the definition of insanity,” Kater admitted, “but I think experiment needs to involve some of that.”

I watched the exchange via Skype, with more interest than I’d have watched the Oscars with. Someday, I hope, I’ll be able to weigh in on such a debate, despite working as a theorist. Someday, I’ll have partnered with enough experimentalists to develop insight.

I’m partnering with Jonathan and Kater on an experiment that coauthors and I proposed in a paper blogged about here. The experiment centers on an uncertainty relation, an inequality of the sort immortalized by Werner Heisenberg in 1927. Uncertainty relations imply that, if you measure a quantum particle’s position, the particle’s momentum ceases to have a well-defined value. If you measure the momentum, the particle ceases to have a well-defined position. Our uncertainty relation involves weak measurements. Weakly measuring a particle’s position doesn’t disturb the momentum much and vice versa. We can interpret the uncertainty in information-processing terms, because we cast the inequality in terms of entropies. Entropies, described here, are functions that quantify how efficiently we can process information, such as by compressing data. Jonathan and Kater are checking our inequality, and exploring its implications, with a superconducting qubit.

With chip

I had too little experience to side with Jonathan or with Kater. So I watched, and I contemplated how their opinions would sound if expressed about theory. Do I try one strategy again and again, hoping to change my results without changing my approach? 

At the Perimeter Institute for Theoretical Physics, Masters students had to swallow half-a-year of course material in weeks. I questioned whether I’d ever understand some of the material. But some of that material resurfaced during my PhD. Again, I attended lectures about Einstein’s theory of general relativity. Again, I worked problems about observers in free-fall. Again, I calculated covariant derivatives. The material sank in. I decided never to question, again, whether I could understand a concept. I might not understand a concept today, or tomorrow, or next week. But if I dedicate enough time and effort, I chose to believe, I’ll learn.

My decision rested on experience and on classes, taught by educational psychologists, that I’d taken in college. I’d studied how brains change during learning and how breaks enhance the changes. Sense, I thought, underlay my decision—though expecting outcomes to change, while strategies remain static, sounds insane.

Old cover

Does sense underlie Kater’s suggestion, likened to insanity, to keep fabricating amplifiers as before? He’s expressed cynicism many times during our collaboration: Experiment needs to involve some insanity. The experiment probably won’t work for a long time. Plenty more things will likely break. 

Jonathan and I agree with him. Experiments have a reputation for breaking, and Kater has a reputation for knowing experiments. Yet Jonathan—with professionalism and politeness—remains optimistic that other methods will prevail, that we’ll meet our goals early. I hope that Jonathan remains optimistic, and I fancy that Kater hopes, too. He prophesies gloom with a quarter of a smile, and his record speaks against him: A few months ago, I met a theorist who’d collaborated with Kater years before. The theorist marveled at the speed with which Kater had operated. A theorist would propose an experiment, and boom—the proposal would work.

Sea monsters

Perhaps luck smiled upon the implementation. But luck dovetails with the sense that underlies Kater’s opinion: Experiments involve factors that you can’t control. Implement a protocol once, and it might fail because the temperature has risen too high. Implement the protocol again, and it might fail because a truck drove by your building, vibrating the tabletop. Implement the protocol again, and it might fail because you bumped into a knob. Implement the protocol a fourth time, and it might succeed. If you repeat a protocol many times, your environment might change, changing your results.

Sense underlies also Jonathan’s objections to Kater’s opinions. We boost our chances of succeeding if we keep trying. We derive energy to keep trying from creativity and optimism. So rebelling against our PhD supervisors’ sense is sensible. I wondered, watching the Skype conversation, whether Kater the student had objected to prophesies of doom as Jonathan did. Kater exudes the soberness of a tenured professor but the irreverence of a Californian who wears his hair slightly long and who tattooed his wedding band on. Science thrives on the soberness and the irreverence.

Green cover

Who won Jonathan and Kater’s argument? Both, I think. Last week, they reported having fabricated amplifiers that work. The lab followed a protocol similar to their old one, but with more conscientiousness. 

I’m looking forward to watching who wins the debate about how long the rest of the experiment takes. Either way, check out Jonathan’s talk about our experiment if you attend the American Physical Society’s March Meeting. Jonathan will speak on Thursday, March 5, at 12:03, in room 106. Also, keep an eye out for our paper—which will debut once Jonathan coaxes the amplifier into synching with his qubit.


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Approximating Hamiltonian dynamics with the Nyström method



Alessandro Rudi1, Leonard Wossnig2,3, Carlo Ciliberto2, Andrea Rocchetto2,4,5, Massimiliano Pontil6, and Simone Severini2

1INRIA – Sierra project team, Paris, France
2Department of Computer Science, University College London, London, United Kingdom
3Rahko Ltd., London, United Kingdom
4Department of Computer Science, University of Texas at Austin, Austin, United States
5Department of Computer Science, University of Oxford, Oxford, United Kingdom
6Computational Statistics and Machine Learning, IIT, Genoa, Italy

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Simulating the time-evolution of quantum mechanical systems is BQP-hard and expected to be one of the foremost applications of quantum computers. We consider classical algorithms for the approximation of Hamiltonian dynamics using subsampling methods from randomized numerical linear algebra. We derive a simulation technique whose runtime scales polynomially in the number of qubits and the Frobenius norm of the Hamiltonian. As an immediate application, we show that sample based quantum simulation, a type of evolution where the Hamiltonian is a density matrix, can be efficiently classically simulated under specific structural conditions. Our main technical contribution is a randomized algorithm for approximating Hermitian matrix exponentials. The proof leverages a low-rank, symmetric approximation via the Nyström method. Our results suggest that under strong sampling assumptions there exist classical poly-logarithmic time simulations of quantum computations.

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Cited by

[1] Ewin Tang, “Quantum-inspired classical algorithms for principal component analysis and supervised clustering”, arXiv:1811.00414.

[2] Juan A. Acebron, “A Monte Carlo method for computing the action of a matrix exponential on a vector”, arXiv:1904.12759.

[3] Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chunhao Wang, “Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning”, arXiv:1910.06151.

The above citations are from SAO/NASA ADS (last updated successfully 2020-02-20 15:40:36). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-02-20 15:40:34: Could not fetch cited-by data for 10.22331/q-2020-02-20-234 from Crossref. This is normal if the DOI was registered recently.


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Extension of the Alberti-Ulhmann criterion beyond qubit dichotomies



Michele Dall’Arno1,2, Francesco Buscemi3, and Valerio Scarani1,4

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore
2Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, Tokyo 169-8050, Japan
3Graduate School of Informatics, Nagoya University, Chikusa-ku, 464-8601 Nagoya, Japan
4Department of Physics, National University of Singapore, 2 Science Drive 3, 117542, Singapore

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The Alberti-Ulhmann criterion states that any given qubit dichotomy can be transformed into any other given qubit dichotomy by a quantum channel if and only if the testing region of the former dichotomy includes the testing region of the latter dichotomy. Here, we generalize the Alberti-Ulhmann criterion to the case of arbitrary number of qubit or qutrit states. We also derive an analogous result for the case of qubit or qutrit measurements with arbitrary number of elements. We demonstrate the possibility of applying our criterion in a semi-device independent way.

As soon as entanglement was recognised as a resource, theorists started studying the interconversions properties of this resource. The most famous such question is: given N copies of a state rho, how many copies N’ of the state rho’ can one obtain with local operations and classical communication? This question led to the definition of entanglement of formation (rho is the maximally entangled state), of distillation (rho’ is the maximally entangled state), to the discovery of inequivalent entanglement classes for multipartite systems… The amount of literature on this question is enormous.

Very little however is known about a different problem, the one we consider here. The question is whether a pair of states (rho,sigma) can be converted into another pair of states (rho’,sigma’). This question does not need to refer to entanglement: in fact, here we don’t consider composite systems, and consequently we don’t restrict the possible operations. A very simple answer would be the one that holds for classical probability distributions: Pair 1 can be converted into Pair 2, if all the statistics that can be observed with Pair 2 can also be observed with Pair 1. This conveys the idea that Pair 1 can do all that Pair 2 can do, and possibly more. This answer holds for two states of qubits (Alberti and Uhlmann, 1980), but counter-examples are known already when Pair 1 comprises qutrit states. In this paper, we prove that the classical-like characterisation still holds when Pair 1 is generalized to any family of qubit states, as soon as they can all be expressed with real coefficients, and Pair 2 is generalized to any family of qubit or, under certain hypotheses, qutrit, states. We also exploit a duality between states and measurements to present a similar characterisation of measurement devices.

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[28] Michael J. Todd, Minimum-Volume Ellipsoids: Theory and Algorithms, (Cornell University, 2016).

[29] S. Boyd and L. Vandenberghe, Convex Optimization, (Cambridge University Press, 2004).

[30] G. M. D’Ariano, G. Chiribella, and P. Perinotti, Quantum Theory from First Principles: An Informational Approach (Cambridge University Press, 2017).

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